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124 \label{ssec:2-cats} |
124 \label{ssec:2-cats} |
125 Let $\cC$ be a disk-like 2-category. |
125 Let $\cC$ be a disk-like 2-category. |
126 We will construct from $\cC$ a traditional pivotal 2-category. |
126 We will construct from $\cC$ a traditional pivotal 2-category. |
127 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
127 (The ``pivotal" corresponds to our assumption of strong duality for $\cC$.) |
128 |
128 |
129 We will try to describe the construction in such a way the the generalization to $n>2$ is clear, |
129 We will try to describe the construction in such a way that the generalization to $n>2$ is clear, |
130 though this will make the $n=2$ case a little more complicated than necessary. |
130 though this will make the $n=2$ case a little more complicated than necessary. |
131 |
131 |
132 Before proceeding, we must decide whether the 2-morphisms of our |
132 Before proceeding, we must decide whether the 2-morphisms of our |
133 pivotal 2-category are shaped like rectangles or bigons. |
133 pivotal 2-category are shaped like rectangles or bigons. |
134 Each approach has advantages and disadvantages. |
134 Each approach has advantages and disadvantages. |
584 The objects of $A$ are $\cC(pt)$. |
584 The objects of $A$ are $\cC(pt)$. |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
585 The morphisms of $A$, from $x$ to $y$, are $\cC(I; x, y)$ |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
586 ($\cC$ applied to the standard interval with boundary labeled by $x$ and $y$). |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
587 For simplicity we will now assume there is only one object and suppress it from the notation. |
588 |
588 |
589 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\times A\to A$. |
589 A choice of homeomorphism $I\cup I \to I$ induces a chain map $m_2: A\otimes A\to A$. |
590 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
590 We now have two different homeomorphisms $I\cup I\cup I \to I$, but they are isotopic. |
591 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
591 Choose a specific 1-parameter family of homeomorphisms connecting them; this induces |
592 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
592 a degree 1 chain homotopy $m_3:A\ot A\ot A\to A$. |
593 Proceeding in this way we define the rest of the $m_i$'s. |
593 Proceeding in this way we define the rest of the $m_i$'s. |
594 It is straightforward to verify that they satisfy the necessary identities. |
594 It is straightforward to verify that they satisfy the necessary identities. |